derivation of flip-flop input equations and state...
TRANSCRIPT
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ECE 331 – Digital System Design
Derivation of Flip-Flop Input Equationsand
State Assignment
(Lecture #24)
The slides included herein were taken from the materials accompanying
Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.
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Spring 2011 ECE 331 - Digital System Design 2
Sequential Circuit Design1. Understand specifications
2. Draw state graph (to describe state machine behavior)
3. Construct state table (from state graph)
4. Perform state reduction (if necessary)
5. Assign a binary value to each state (state assignment)
6. Create state transition table
7. Select type of Flip-Flop to use
8. Derive Flip-Flop input equations and FSM output equation(s)
9. Draw circuit diagram
![Page 3: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/3.jpg)
Spring 2011 ECE 331 - Digital System Design 3
Derivation of Flip-Flop Input Equations
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Spring 2011 ECE 331 - Digital System Design 4
Derivation of FF Input Equations
Example #2:
Derive the Flip-Flop input equations for the FSM described by the following state table.
Assume that JK Flip-Flops are used in the design.
Excitation Table:
Q Q+ J K
0 0 0 x
0 1 1 x
1 0 x 1
1 1 x 0
![Page 5: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/5.jpg)
Spring 2011 ECE 331 - Digital System Design 5
Example #2: FF Input Equations
State Table
Present Next State Output
State X = 0 X = 1 X = 0 X = 1
S0 S1 S2 0 1
S1 S2 S3 0 0
S2 S3 S0 1 0
S3 S0 S1 0 1
![Page 6: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/6.jpg)
Spring 2011 ECE 331 - Digital System Design 6
Example #2: FF Input Equations
1. Assign a binary value to each state.2. Construct the state transition table.
A+B+ JAKA JBKB
AB X = 0 X = 1 X = 0 X = 1 X = 0 X = 1
00
01
10
11
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Spring 2011 ECE 331 - Digital System Design 7
Example #2: FF Input Equations
3. Construct K-maps for Flip-Flop inputs.4. Derive the minimized FF input equation.
JA = KA =
![Page 8: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/8.jpg)
Spring 2011 ECE 331 - Digital System Design 8
Example #2: FF Input Equations
3. Construct K-maps for Flip-Flop inputs.4. Derive the minimized FF input equation.
JB = KB =
![Page 9: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/9.jpg)
Spring 2011 ECE 331 - Digital System Design 9
Derivation of FF Input Equations
Example #3:
Derive the Flip-Flop input equations for the FSM described by the following state table.
Assume that SR Flip-Flops are used in the design.
Excitation Table:
Q Q+ S R
0 0 0 x
0 1 1 0
1 0 0 1
1 1 x 0
![Page 10: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/10.jpg)
Spring 2011 ECE 331 - Digital System Design 10
Example #3: FF Input Equations
State Table
Present Next State Output
State X = 0 X = 1 X = 0 X = 1
S0 S1 S2 0 1
S1 S2 S3 0 0
S2 S3 S0 1 0
S3 S0 S1 0 1
![Page 11: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/11.jpg)
Spring 2011 ECE 331 - Digital System Design 11
Example #3: FF Input Equations
1. Assign a binary value to each state.2. Construct the state transition table.
A+B+ SARA SBRB
AB X = 0 X = 1 X = 0 X = 1 X = 0 X = 1
00
01
10
11
![Page 12: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/12.jpg)
Spring 2011 ECE 331 - Digital System Design 12
Example #3: FF Input Equations
3. Construct K-maps for Flip-Flop inputs.4. Derive the minimized FF input equation.
SA = RA =
![Page 13: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/13.jpg)
Spring 2011 ECE 331 - Digital System Design 13
Example #3: FF Input Equations
3. Construct K-maps for Flip-Flop inputs.4. Derive the minimized FF input equation.
SB = RB =
![Page 14: Derivation of Flip-Flop Input Equations and State …ece.gmu.edu/~clorie/Spring11/ECE-331/Lectures/Lecture_24.pdfDerivation of Flip-Flop Input Equations and State Assignment ... Derivation](https://reader030.vdocuments.us/reader030/viewer/2022012401/5aa4cfed7f8b9a185d8c83c0/html5/thumbnails/14.jpg)
Spring 2011 ECE 331 - Digital System Design 14
State Assignment
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Spring 2011 ECE 331 - Digital System Design 15
State Assignment
● After the number of states in the state table has been reduced …
● A binary value must be assigned to each of the states.
– State assignment (or state encoding)
– Binary value = state of Flip-Flops
● The cost of the logic required to realize the FSM is dependent on the state assignment.
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Spring 2011 ECE 331 - Digital System Design 16
State Assignment
● Given: A FSM with three states.
● Requires: Two Flip-Flops (A and B)
– Can implement a maximum of four states.
● There are 4 x 3 x 2 = 24 possible state assignments.
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Spring 2011 ECE 331 - Digital System Design 17
State Assignment
● For a FSM realized using symmetrical Flip-Flops (i.e. JK and SR)
– 3 unique state assignments for 3-state FSM
– 3 unique state assignments for 4-state FSM
Binary GrayCode
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Spring 2011 ECE 331 - Digital System Design 18
State Assignment
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Spring 2011 ECE 331 - Digital System Design 19
Guidelines for State Assignment
The author provides a set of guidelines by which the optimal state assignment can be selected.
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Spring 2011 ECE 331 - Digital System Design 20
One-Hot State Assignment
● Sometimes, reducing the next-state logic is more important than reducing the number of Flip-Flops.
● One-hot state assignment may result in minimal next-state logic.
– Uses one Flip-Flop per state.
– Exactly one Flip-Flop is set to 1 for each state.
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Spring 2011 ECE 331 - Digital System Design 21
Example: State Assignments
For a 4-state FSM, three possible state assignments are:
State Binary Gray-code One-hot
S0 00 00 0001
S1 01 01 0010
S2 10 11 0100
S3 11 10 1000
# of FF 2 2 4
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Spring 2011 ECE 331 - Digital System Design 22
Example: State Assignments
Binary state-assignment:
Present Next State
State X = 0 X = 1
S0 S1 S2
S1 S2 S3
S2 S3 S0
S3 S0 S1
A+B+
AB X = 0 X = 1
00 01 10
01 10 11
10 11 00
11 00 01
State Transition TableState Table
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Spring 2011 ECE 331 - Digital System Design 23
Example: State Assignments
Gray-code state-assignment:
Present Next State
State X = 0 X = 1
S0 S1 S2
S1 S2 S3
S2 S3 S0
S3 S0 S1
A+B+
AB X = 0 X = 1
00 01 11
01 11 10
11 10 00
10 00 01
State Transition TableState Table
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Spring 2011 ECE 331 - Digital System Design 24
Example: State Assignments
One-hot state-assignment:
Present Next State
State X = 0 X = 1
S0 S1 S2
S1 S2 S3
S2 S3 S0
S3 S0 S1
A+B+C+D+
ABCD X = 0 X = 1
0001 0010 0100
0010 0100 1000
0100 1000 0001
1000 0001 0010
State Transition TableState Table
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Spring 2011 ECE 331 - Digital System Design 25
Questions?